Positive Solutions for Second–Order m–Point Boundary Value Problems on Time Scales

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb T}$$\end{document} be a time scale such that 0, T ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb T}$$\end{document}. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m–point boundary value problem on time scales \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{array}{*{20}c} {{u^{{\Delta \nabla }} {\left( t \right)} + a{\left( t \right)}f{\left( {u{\left( t \right)}} \right)} = 0,t \in {\left( {0,T} \right)},}} & {{\beta u{\left( 0 \right)} - \gamma u^{\Delta } {\left( 0 \right)},u{\left( T \right)} - {\sum\limits_{i = 1}^{m - 2} {a_{i} u{\left( {\xi _{i} } \right)}} } = b,m \geqslant 3,}} \\ \end{array} $$\end{document} where a ∈ Cld ((0, T), [0,∞)), f ∈ Cld ([0,∞) × [0,∞), [0,∞)), β, γ ∈ [0,∞), ξi ∈ (0, ρ(T)), b, ai ∈ (0,∞) (for i = 1, . . . ,m− 2) are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b > 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb T}$$\end{document} = ℝ) and difference equation (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb T}$$\end{document} = ℤ).